Mr Barton

1
Mr Bartons Maths Notes

• Trigonometry
• 3. 3D

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images!
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3. 3D Trigonometry

• The Secret to Solving 3D Trigonometry Problems
• 3D Trigonometry is just the same as
  bog-standard, flat, normal trigonometry
• All we need are the skills we learnt in the last
  two sections
• 1. Pythagoras
• 2. Sin, Cos and Tan
• The only difference is that is a little bit
  harder to spot the right-angled triangles
• But once you spot them
• Draw them out flat
• Label your sides
• Fill in the information that you do know
• Work out what you dont in the usual way!
• And if you can do that, then you will be able to
  tick another pretty tricky topic off your list!
• Please Remember You need a right-angled triangle
  to be able to use either Pythagoras or Sin, Cos
  and Tan and I promise that will be the last time
  I say it!

3
Example 1
The diagram below shows a record breaking wedge
of Cheddar Cheese in which rectangle PQRS is
perpendicular (at 900 to) to rectangle RSTU. The
distances are shown on the diagram. Calculate
(a) The distance QT (b) The angle QTR
P
S
Q
2.5 m
R
T
4.9 m
7.8 m
U
4
Working out the answer (a)
The first thing we need to figure out is what we
are actually trying to work out! We need the line
QT
P
S
Q
2.5 m
R
T
4.9 m
7.8 m
U
Now, as I said, the key to this is spotting the
right-angled triangles Well, I can see a nice
one TQR. That contains the length we want, and
we already know how long QR is So now all we
need to do is work out length TR
5
Working Out TR
P
Okay, if you look carefully, you should be able
to see a right-angled triangle on the base of
this wedge of cheese Its the triangle TRU
S
Q
2.5 m
R
T
T
4.9 m
7.8 m
c
a
?
U
4.9 m
Well, we have two sides and we want to work out
the Hypotenuse This looks like a job for
Pythagoras!
R
U
7.8 m
b
1. Label the sides 2. Use the formula 3. Put in
the numbers
6
Working Out TQ
P
S
Okay, so now we have all we need to be able to
calculate TQ. Just make sure you draw the correct
right-angled triangle!
Q
2.5 m
R
T
9.21 m
Q
4.9 m
7.8 m
c
?
U
2.5 m
b
T
R
9.21 m
Once again, we have two sides and we want to work
out the Hypotenuse This looks like a job for
Pythagoras!
a
1. Label the sides 2. Use the formula 3. Put in
the numbers
7
Working out the answer (b)
P
Again, we must be sure we know what angle the
question wants us to find! I have marked angle
QTR on the diagram So now we draw our
right-angled triangle
S
Q
2.5 m
9.54 m
R
T
9.21 m
4.9 m
7.8 m
U
Q
H
O
9.54 m
To calculate the size on an angle, we must use
either sin, cos or tan, which means first we must
label our sides! Now, because we actually know
all three lengths, we can choose! Im going for
tan!
2.5 m
T
R
9.21 m
A
Tan ? o a
Tan ? 0.27144
15.20 (1dp)
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Example 2
The diagram below shows a plan of a tent that I
am trying to erect before the rain comes. OP is a
vertical pole, and O is at the very centre of the
rectangle QRST. The lengths and angles are as
shown on the diagram. Calculate the height of the
vertical pole OP.
P
Q
O
48o
R
T
5 m
12 m
S
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1. Working Out OT
T
You should be able to see that if we can work out
OT, we will then have a right-angled triangle
which will give us OP! So, to get started we need
to use the base of the rectangle Well, OT is
half way along the line TR line, so it must be
6.5m
?
c
a
5 m
R
S
12 m
b
P
H
O
?
480
T
O
2. Working Out OP
6.5 m
A
And now we have a right-angled triangle where we
know one length (TR), and we know one angle
(OTP) so we can work out any side using a bit of
sin, cos or tan!
o Tan ? x a
7.22 m (2dp)
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• Good luck with your revision!